1. He gave us a piece of paper.

Question

2. He gave each of us a piece of paper.
3. He gave us the paper.

Does #1 mean #2?
What about #3? Is #3 correct?

Answer 1

The first sentence could be interpreted that he gave the group (us) one piece of paper. The second sentence makes it clear that each person was given a piece of paper. The third sentence isn't clear about how much paper he gave us. It could mean the same as either # 1 or # 2.

Answer 2

Thank you for your explanation.

1. Write down your name on the paper.
2. Write down your name on a paper.
3. Write down your name on a piece of paper.
4. Write down your name on the piece of paper.

I think all are correct except #2? Am I right?

Answer 3

# 2 could be correct if the only thing the other person wants is to see your name.

We also don't need the word "down" in these sentences. "Write your name on the paper" is the usual expression.